Re: order of magnitude of an inverse

In article <1155553733.732937.40070@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
oercim <oercim@xxxxxxxxx> wrote:

Robert Israel wrote:
In article <1155460729.954548.143870@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
oercim <oercim@xxxxxxxxx> wrote:
If A = O(n) + aO(sqrt(n)) , then what is the
order of
magnitude of A (-1). Here a is unknown, it can be a scalar or a
function of n

Not determined.

Sorry, I wasn't clear. With A (-1) , I meant A^(-1).


What is A? A function of n, I guess, but are the values
numbers, matrices, what? Also, I presume you're talking about
integers n -> infinity?

A is a matrix. Yes, i am talking about as n goes to
infinity.


If a is constant (i.e. not dependent on n), then
O(n) + a O(sqrt(n)) = O(n). All this means is that there are
some constants N and K such that |A| <= K n for n > N (if
A is a matrix, this means all |A_{ij}| <= K n for n > N).
If a is a function of n, then without further information
on a there's nothing you can say.

Can't we say about the order of magnitude in terms
of a? (for
example:
A^(-1)=a^(-1)O(n^(-1)).


If A = O(n), there's no guarantee that A^(-1) exists: A could
be 0. If A is a number and nonzero, A^(-1) = Omega(1/n),
i.e. there exist constants N and K > 0 such that
|A^(-1)| >= K/n for n > N.
Assume A exists. I think A^(-1) should be
O(n^(-1))+a^(-1)O(n^(-1/2)). And let say if a=1/n then
A^(-1)=O(n^(-1))+a^(-1)O(n^(-1/2))

=O(n^(-1))+O(n^(1/2))
=O(n^(1/2))

I assume you mean assume A^(-1) exists. Consider e.g.
the 2 x 2 matrices
[use fixed-width font]

[ n 0 ]
A = [ 0 f(n) ]

[ 1/n 0 ]
A^(-1) = [ 0 1/f(n) ]

with f(n) -> 0 as rapidly as you want as n -> infty.
Then A = O(n), but unless you have a _lower_ bound on how
rapidly f(n) -> 0, there's no way to get a bound on
A^(-1).

Robert Israel israel@xxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada

.

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